What Are Recursive Sequences?

$\textbf{Dividing Recurrence relations into types}$

A recurrence relation is an expression that relates each term of a sequence to it's previous terms. For example $a_n = 4a_{n-1}+3a_{n-2}-6$. Here $n^{th}$ term's recursive formula, dependence on 2 previous terms is given.

Recurrence relations which have such an expression containing only the previous terms raised to power 1, with some coefficient, and contain no constants or variables other than previous terms in the recurrence are called as $\text{Homogeneous Recurrence relations}$.

Some examples of homogeneous recurrence relations are as follows

$a_n=2a_{n-1}$... (See,this will be a GP)

$a_n=a_{n-1}+6a_{n-2}$

And the examples of recurrence relations which are not homogeneous, are

$a_n=a_{n-1}+6$

$a_{n}=a_{n-1}+3a_{n-2}+n$

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Here is an exercise...